Implications

We have the following indirect implication of form equivalence classes:

133 \(\Rightarrow\) 223
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
133 \(\Rightarrow\) 63 Dedekind-Endlichkeit und Wohlordenbarkeit, Brunner, N. 1982a, Monatsh. Math.
63 \(\Rightarrow\) 70 clear
70 \(\Rightarrow\) 206 clear
206 \(\Rightarrow\) 223 clear

Here are the links and statements of the form equivalence classes referenced above:

Howard-Rubin Number Statement
133:  

Every set is either well orderable or has an infinite amorphous subset.

63:

\(SPI\): Weak ultrafilter principle: Every infinite set has a non-trivial ultrafilter.
Jech [1973b], p 172 prob 8.5.

70:

There is a non-trivial ultrafilter on \(\omega\). Jech [1973b], prob 5.24.

206:

The existence of a non-principal ultrafilter: There exists an infinite set \(X\) and a non-principal ultrafilter on \(X\).

223:

There is an infinite set \(X\) and a non-principal measure on \(\cal P(X)\).

Comment:

Back